There are a vast difference in qualities of screens available on the market. If you go for the cheapest option,(eg glass replacement) then you will likely encounter problems with the screens.(eg: overheating and battery draining)You want to look for someone who offers a 6-12 month warranty on the screen. The length of the warranty speaks volumes about the quality which is being used.
We have Genuine LCD screen with 12 month warranty, but we can also do after market (copy)screen with 3 month warranty if customers requested and for iPhone screens, We may have 1 out of every 100 which are returned under warranty.
We find most people base their decision on price. Where they would be much better off in the long run basing it on length of warranty.